SO(3) Special Othorgonal Group

[logarithmic]

For the special orthogonal group SO(3), with G-set R^3, and the usual G-action, we choose x in R^3 not equal to 0. Then the stabilizer of x (set of all the transformations in SO(3) that doesn't change x) is all the rotations about the axis produced by x (and -x). Can someone explain why the stabilizer of x is isomorphic to SO(2)? The notes I have just writes this as if it should be obvious. Am I missing something? Also, just to make sure I'm understanding this right, is the geometric interpretation for SO(2) the group of all rotations by any angle in R^2?

 

[Dick]

The set of rotations around an axis x in R^3 is isomorphic to the set of rotations in the plane orthogonal to x. And, yes, SO(2) is the set of rotations of a plane.